Trying to make a figure eight
You do not draw the circle and then cut, you just draw the arc. 

This can be easily achieved by starting with 1 point (vertex). 

Then select that vertex in edit mode and place the (before thought of as useless) 3D cursor in the center of the rotation and click spin inside the mesh tools. in the popup choose the angle and be happy you got an arc ;)

Spin always extrudes the selection, so if you want to add another arc (perhaps with different angle and rotation point) at the end of you new arc, just select the start vertex.

Note: Make sure you are in orthogonal view, or else you might not end up at the same depth you started with ;) You switch between orthogonal and perspective view by pressing NUM5.